Shape oscillation and transition from tank-treading to tumbling of a nonspherical capsule in shear flow*

Dynamics of a nonspherical capsule in shear flow is studied using a front-tracking method. The capsule is modeled as a liquid drop enclosed by a thin hyperelastic membrane following (i) the strain-softening neo-Hookean law, and (ii) the strain- hardening Skalak law. The undeformed shape of the capsule is an oblate spheroid with aspect ratio $\alpha$ (ratio of the minor- to-major axis) varying from 1 to 0.6. The viscosity ratio $\lambda$ of the interior to exterior liquid varies from 1 to 20. Unlike for an initially spherical capsule, the Taylor deformation parameter $D$ and the orientation of the nonspherical capsule oscillate in time. We present phase diagrams in terms of dimensionless shear rate (Capillary number, $Ca$), viscosity ratio $\lambda$ and the aspect ratio $\alpha$ to describe regimes of various modes of capsule orientation dynamics. At $\lambda \approx 1$, a swinging mode is observed in which the orientation oscillates about a mean value. At higher $\lambda$, a breathing mode is observed in which the fails to make a complete rotation, and $D$ changes significantly over one cycle. When $\lambda$ is further increased, the full tumbling motion is established. Numerical results are compared with the theory of Keller and Skalak, JFM (1982), vol. 120, and Skotheim and Secomb, PRL (2007), vol. 98.